Graphing Integers

Graphing on the coordinate plane is a way to visualize relationships between two quantities. René Descartes was the French mathematician and philosopher who discovered this unique way to combine algebra and geometry. He recognized that by drawing two perpendicular number lines called axes, and labeling them with positive and negative numbers, he could locate any point on the plane with x- and y-coordinates. The diagram below shows the different parts of the coordinate plane

The point where the x-axis and y-axis intersect is called the origin. Notice that x-values to the right of the y-axis are positive and x-values to the left of the y-axis are negative. Similarly, y-values above the x-axis are positive and y-values below the x-axis are negative. The axes divide the plane into four quadrants which are numbered starting in the upper right hand quadrant going counterclockwise as I, II, III, and IV.

An ordered pair of numbers is used to locate any point on the plane. An ordered pair is enclosed in a pair of parentheses with the first number representing a location on the x-axis, the x-coordinate, and the second number representing a location on the y-axis, the y-coordinate. To locate a point on the coordinate plane, do the following:

(1)

Start at the origin.

(2)

The first number is the x-coordinate. If it is positive, move to the right the appropriate number of units. If it is negative, move to the left the appropriate number of units.

(3)

The second number is the y-coordinate. If it is positive, move up the appropriate number of units . If it is negative, move down the appropriate number of units.

Look at the coordinate grid below. The ordered pair for point A is (3, -2). To locate point A , we move three units to the right and two units down. Also shown on the coordinate plane are points B (-3, 5), C (-1, -4), D (0, -3), E (2,4) and F (4,0).

Notice that points in quadrant I have both a positive x- and y-coordinate. Points in quadrant II have a negative x-coordinate and a positive y-coordinate. Points in quadrant III have both a negative x- and y-coordinate, and points in quadrant IV have a positive x-coordinate and a negative y-coordinate.

One way to represent relationships between pairs of numbers is through the use of an equation like x  2 = y. A second way to represent that same relationship is through the use of a function table like the one below.

x

x  2 = y

y

(x, y)

-3

-3  2 = -5

-5

(-3, -5)

-2

-2  2 = -4

-4

(-2, -4)

-1

-1  2 = -3

-3

(-1, -3)

0

0  2 = -2

-2

(0, -2)

1

1  2 = -1

-1

(1, -1)

2

2  2 = 0

0

(2, 0)

3

3  2 = 1

1

(3, 1)

This table can then be used to create the graph of the relationship. By plotting the points from the table above, we can see the relationship which exists between the points which satisfy the equation x  2 = y.

We can now see that the pattern of dots lies along a straight line. By connecting the dots with a solid line and indicating that it can be extended in both directions, we have the graph of x  2 = y. A linear function is one which when graphed becomes a straight line.